The flax plant has blue flowers and drops itself when broken. Like most plants, it can be broken easily with one hit of a hand.
There are three different flax-related items: the flax plant which grows wild, the flax seeds which are crafted from the flax plant, and the flax item that is harvested from farming. Only the flax item can be crafted into string, not the plant directly.
Each flax plant can be crafted into two flax seeds. The seeds can be planted on tilled ground.
Once they are grown, harvesting the plants will yield more seeds and flax items. Each of these flax items can then be crafted into one piece of string.
The only use and motivation for farming flax is the possibility to craft it into string. Unlike other mod flowers, flax can not be crafted into dyes.
While it may not seem profitable to farm flax at first glance, string is a valuable commodity: many Wood-elf, Galadhrim or Lindon Smiths buy string for two coins each, allowing you to get two coins per flax. To make a lot of money off flax, all you have to do is find and mark several elven smiths that buy string, then make a large flax farm. When you harvest your flax, convert it to string, then sell it to one of the elven smiths. This also allows you to make large amounts of money from bones, about two coins for each heap of bone meal.
Before Public Beta 24, one flax item gave two strings when being crafted.